Precalculus 1: basic notions

A total of 52 hours of lectures

Due to the size of precalculus, it is divided into four parts. This page describes the first of those four parts.

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Prerequisites

High-school maths, mainly arithmetics, some trigonometry

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Curriculum

Make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

Precalculus: basic notions

Get the outline

A detailed list of all the lectures in part 1 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

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Course Objectives & Outcomes

How to solve problems in chosen Precalculus topics (illustrated with 236 solved problems) and why these methods work.

The basic notions will be defined in a more abstract way, and you will get an insight into their place in Calculus and some other branches of university maths.

Arithmetic with basic rules (associativity and commutativity of addition and multiplication, distributivity) illustrated for positive integers.

Basic concepts related to functions (domain, range, graph, surjection, injection, bijection, inverse, composition) and how to work with them.

Geometrical operations on graphs of functions: translations, reflections, shrinking, dilating. Piecewise functions.

Equations of straight lines in the plane: slope and intercept; first glimpse into derivative as the slope of the tangent line, and link to monotonicity.

Logic: symbols and rules (tautologies) used for creating mathematical statements, definitions, and proofs.

Relations: RST (equivalence) relations, order relations, functions as relations.

Various types of proofs: direct, indirect, by contradiction, induction proof, with some examples.

Decimal expansion of rational and irrational numbers. Density of Q and R\Q in R.

You will also get a solid background for any future studies in Real Analysis and other courses in university mathematics.

Repetition of chosen aspects of high school mathematics (basic notions as numbers, functions, sets, equations, and inequalities).

By the end of this course you will be able to understand all the elements in the epsilon-delta definition of limit, both symbols and the content.

Basic terminology for equations and inequalities, with examples of linear equations and inequalities, with and without absolute value.

Functions: monotone (increasing, decreasing), bounded, even, odd; extremum: maximum, minimum, both local and global.

Absolute value and its role in computing distances, with geometrical illustrations and functional approach.

Introduction to sequences and series; short notation for sum (Sigma) and product (Pi); arithmetic, geometric, and harmonic progressions.

Set theory: union, intersection, complement, set difference; some important rules and notation.

Necessary and sufficient conditions: definition and examples.

Building blocks of mathematical theories: axioms, definitions, theorems, propositions, lemmas, corollaries, etc.

Cardinality of sets: finite sets, N, Z, Q, R; countable and uncountable sets.

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Lecturer mathematics

The bottom blocks in a pyramid are removed at the peril of the top, and the educational establishment has zealously been doing just that. The crying gap between the school graduate and mathematical analysis at university was first masked with something called calculus, and then, the gap always growing, with an additional repair putty by the apt name of pre-calculus, to be quietly injected between the devalued academia and the ruins of the common school. Had it not been for the able and enthusiastic colleagues, like Hania, masters at educational restoration, the neglected structure would have long collapsed. Indeed, Hania not only restores the foundations, but improves them, giving the traditional material a very much needed face-lift. Excelling university teachers seldom touch school mathematics, but when they do, the students get to see that a strangled school subject can in fact be hiding a door to wonderland.

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Lecturer mathematics

This is a detailed exposition of the many basic things that have the habit of falling between the curricula of the school and those of the university. The course can readily be taken by interested school students as well as by the university students in need of grounding what they in principle should know, though perhaps do not in practice. Insufficient mathematical foundations are an old problem at university, and Hania’s present course – comprehensive and immediately rewarding – addresses that problem exquisitely.